Effect of Temperature on Dry Cell Life Span:
A Case Study
R. Radharamanan 1 and Jeng-Nan Juang 2
Abstract – In this paper, the life of dry cells using a flashlight at different temperatures were tested. The dry cells
used were heavy-duty (non-alkaline) batteries using zinc and carbon electrodes. The objective is to see whether a
dry cell’s life span is affected by temperature because the dry cell relies on a chemical reaction to generate its
energy, and typically, as temperature increases, the rate of chemical reaction also increases. The batteries were
subjected to a specific temperature, 72oF and 170oF, for 15 minutes and then ran until the light emitted by the
flashlight was determined not visible anymore. Once the tests were completed, the data were analyzed and the
results indicated that temperature has a significant effect on the life span of dry cell batteries. Finally, the chemical
reaction’s overall activation energy equation was solved using the average life spans of batteries measured.
Keywords: Life of a dry cell, statistical analysis, Tukey’s test, ANOVA, and temperature effect.
INTRODUCTION
The interest for this experiment came as a result of a desire for practical application. If one is using a piece of
battery-operated equipment in a high-temperature environment (such as the troops using hand-held radios in the in
the deserts of Iraq at >130°F), it would be good to know how long one can expect the equipment to last versus
assuming it would be the same while using it at room temperatures (~72°F). The life spans would be different at
different operating temperature conditions since (a) a battery relies on chemical reaction to make its energy [3] and
(b) temperature affects the rate at which the chemical reaction takes place [8].
In today’s market there are several different types of batteries. For this experiment, AA-sized batteries were chosen.
There are two types of AA batteries: primary and secondary. Non-alkaline and alkaline batteries are known as
primary [4] batteries because after their first use, they must be replaced or discarded. Rechargeable batteries are
referred to as secondary batteries since their chemical makeup allows them to be recharged, and thus, reused. Only
primary batteries were tested in this study.
Primary batteries are very popular in industry, despite their one-time use. The alkaline battery, specifically, though it
must be discarded after its first use, tends to be cheaper, more efficient, and longer lasting than the current
rechargeable batteries available [6]. The focus for this experiment, however, was on the weaker of the two primary
battery types, the non-alkaline battery. The two main advantages that the non-alkaline used to have over the alkaline
battery were that it was much cheaper and was also less likely to leak its internal chemicals [5]. Recently, however,
due to advances in manufacturing and technology, the price and stability of alkaline batteries are closely matching
non-alkaline, thus driving the non-alkaline to extinction.
Mercer University School of Engineering, 1400 Coleman Ave., Macon, GA 31207,
[email protected]
1
Mercer University School of Engineering, 1400 Colman Ave., Macon, GA 31207,
[email protected]
2
2009 ASEE Southeast Section Conference
So why test the weakest battery? Two reasons – it’s cheaper and it takes less time. In a recent study of the life span
of various AA batteries, it was shown that an alkaline battery lasted 4.3 times longer than a heavy duty (nonalkaline) battery [5] for a flashlight test. As for pricing, a simple internet search revealed that when buying in bulk
(1000+ batteries) the cheapest price per battery for non-alkaline was 8¢ /battery, whereas the cheapest for alkaline
was 11¢ /battery.
As stated previously, the driving force behind a battery is the internal chemical reaction taking place. The batteries
tested undergo what is called an oxidation-reduction reaction. To better understand this reaction, look at Figure 1
[10].
In Figure 1, it is seen that a battery has a positive terminal and a negative terminal. The positive terminal functions
as the electric focal point for the carbon rod, which serves to carry electrons to the manganese dioxide mix. The
negative terminal serves as the electric focal point for the zinc. Ultimately, a battery functions by allowing electrons
to flow from the negative terminal to the positive terminal, as shown in Figure 2 [7].
Figure 1: Battery with positive and negative terminals
2009 ASEE Southeast Section Conference
Figure 2: Flow of electrons from negative to positive terminal
The driving force behind the electrons ‘wanting to go’ from one terminal to the other is from these two halfreactions:
Zn(s) Î Zn2+(aq) + 2 e-
(1)
MnO2(s) + H2O(l) + e- Î MnO(OH)(s) + OH-(aq)
(2)
Equation (1) is the oxidation reaction because zinc is oxidized from Zn0 to Zn2+. Equation (2) is the reduction
reaction because Mn4+ is reduced to Mn3+. From Figure 1, it is seen that there is an electrolyte separator between the
zinc and the manganese dioxide, otherwise a reaction would take place internally between the solids and chemicals
and the battery would quickly lose all its electrical power. Equation (2) can only take place if there is a flow of
electrons to come into contact with the MnO2, and the separator prevents this from occurring. The only way for this
to occur is to run a wire between the two electrodes (Fig. 2).
As for measuring the strength of the battery, voltage is the battery’s driving force to push electrons through a circuit.
It is the change in potential energy when a reaction occurs. Equations (1) and (2) have specific voltages associated
with each of them and when combined, equation (3) is the overall chemical reaction, where the voltages add up to
1.5V (the standard voltage for an AA battery):
8MnO2 + 4Zn + ZnCl2 + 9H2O Î 8MnOOH + ZnCl2.4ZnO.5H2O
(3)
Understanding how a battery works internally, now one can briefly analyze a chemical reaction’s temperature
dependence. Temperature is the measure of a substance’s average molecular kinetic energy. Kinetic energy is equal
to ½mv2, where m is the mass of the molecule and v is the average velocity of the molecule based either on the
molecule moving through space (gas or liquid) or vibrating in place (solid). In order for a reaction to take place,
molecules must be interacting and coming into contact to share electrons. The more that molecules move around,
the more these reactions occur. A general rule of thumb is that for every 10°C rise in temperature, the rate of the
reaction doubles [2]. This loosely-followed rule is derived from the Arrhenius equation (eq. 4). This equation can
be rearranged to compare reaction rates at two different temperatures (eq. 5):
k = A e (- Ea/RT)
⎛ E ⎛ 1 1 ⎞⎞
k2
= e⎜⎜ − a ⎜⎜ − ⎟⎟ ⎟⎟
k1
⎝ R ⎝ T2 T1 ⎠ ⎠
2009 ASEE Southeast Section Conference
(4)
(5)
where Ea is the activation energy, the minimum energy required to get the reaction to take place, expressed in J/mol
of reactant; R is the universal gas constant (in this case, 8.3145 J·mol-1·K-1); k is the rate of the reaction; and T is the
temperature. The Ea value in equation (3) is not known otherwise one could directly estimate a battery’s life span
for a given temperature if the life span at the room temperature is known. This can be solved once the data to
estimate the reaction rates at the two temperatures are known.
METHODOLOGY
Having analyzed the chemistry and the potential temperature dependence for the life of a battery, one can look at the
methods of analyzing this phenomenon. The method for testing the life of a battery was to use a flashlight at two
different temperatures and observe it from the initial start time until it was no longer bright enough to be of any use.
This could be a significant source of error since it is difficult to define exactly when the flashlight was ‘dead’. As
the battery’s power decreases, the light of the flashlight does not stay at a constant brightness, but instead becomes
dimmer. So, while the power may be diminishing, its output also diminishes, thus the flashlight could be deemed as
functioning almost indefinitely.
The equipment used for this experiment was two Rayovac® Industrial AA flashlights and 28 GI Super Heavy Duty
Batteries. The two flashlights required two AA batteries for operation, and with seven runs per flashlight at two
separate temperatures, 28 batteries were used. The flashlights were purchased from Lowes® and the batteries were
ordered from www.batteriesandbutter.com.
The flashlights were tested at 72°F and 170°F, which translates to 295°K and 350°K respectively. Conversion to
Kelvin temperatures is needed because equations (4) and (5) require an absolute scale. 295°K was achieved by
testing at room temperature, while 350°K was achieved using a kitchen oven. To ensure that each battery was at the
right temperature prior to testing, both the battery and the flashlight were required to be in the testing environment
for at least 15 minutes. Both the oven and the ambient environments could be controlled by a temperature setting,
but both were closely monitored using a separate thermometer to ensure that the thermal accuracy of at least ± 5°F
was achieved.
As stated previously, judging when a flashlight is ‘dead’ is difficult. A method for better judging this was to set the
flashlight approximately one foot from a white piece of paper in a dark environment. At initial operation, there was
a bright white circle that appeared on the paper. As time increased, the circle became smaller and started to appear
an amber color. The time when the flashlight was declared done was when that amber circle was no longer
reasonably visible.
After observing and collecting the data for each of the 14 runs, statistical tests were carried out to determine if
temperature did affect the life span of the batteries.
RESULTS AND DISCUSSIONS
In order to test the life span of batteries a method known as Analysis of Variance (ANOVA) was used. This type of
testing is used to determine if there is a significant difference in the means of treatments [9]. Single factor
treatments were used with temperature variability at 72oF and 170oF. Fourteen samples were taken, seven at both
72oF and 170oF. The data collected are presented in Table 1. The sample data is described by the linear statistical
model for One-Way-Classification Fixed-Effects Model,
yij = μ + τi + εij { i = 1, 2
{ j = 1, 2
(6)
where μ is the overall mean, τi the ith treatment effect and εij is the random error component [1]. For hypothesis
testing, the model errors are assumed to be normally and independently distributed with mean zero and variance σ2,
and variance is considered constant at all levels [1]. The hypothesis used for testing:
H0: τ72 = τ170 = 0
H1: τi ≠ 0 for at least one i.
2009 ASEE Southeast Section Conference
(7)
The resulting ANOVA One-Way-Classification Fixed-Effects Model is shown in Table 2, where “DF” is Degrees of
Freedom.
Table 1: Sample data collected at ambient and heated conditions
Temperature (oF)
Ambient (72oF)
Heated (170oF)
Battery Life (minutes)
1
73
36
2
75
32
3
76
39
4
76
41
5
85
38
yi.
6
80
37
7
79
31
544
254
yi.
77.71
36.29
Table 2: ANOVA
Source of Variation
SS
DF
MS
F0
F(.05,1,12)
S/NS
Temperature
Error
Total
6007.14
174.86
6182.00
1
12
13
6007.14
14.57
412.25
4.75
S
SS = Sum of Squares; DF = Degrees of Freedom; MS = Mean Square Error; S/NS = Significant/Not Significant.
Factors for the ANOVA model were determined from the following equations and processes [1, 12],
Temperature Sum of Squares (SSTemperature) =
Total Sum of Squares (SST) =
∑y
2
i.
∑ ∑y
2
ij
− y..2 / N
− y..2 / N
(8)
(9)
Error Sum of Squares (SSE) = SST – SSTemperature
(10)
Temperature Mean Square (MSTemperature) = SSTemperature / DF
(11)
Mean Square Error (MSE) = SSE / DF
(12)
F0 = MSTemperature / MSE
(13)
In order to test the null hypothesis the calculated F0 from the ANOVA table is compared to F.05, 1, 12. From the
ANOVA table F0 was found to be 412.25 and from F-Table F.05, 1, 12 was found to be 4.75.
To check the model and affirm the assumption, that errors are normally and independently distributed with constant
variance, residuals were used from the following equations [1, 12]:
Residual (eij) = yij – yi
(14)
yi = 1/n ∑ yi
(15)
where i = 1, 2
Normal Score = ((1 – .5) / N) * 100
(16)
Figures 3, 4, and 5 were made to visually and graphically determine the model adequacy to normality and constant
variance. Figure 3 is the Normal Scores vs. the Residual; Figure 4 is Residuals vs. Fitted Values; Figure 5 is
Residuals vs. Battery Life. These graphs and their significance are further explained below.
2009 ASEE Southeast Section Conference
Normal Scores vs Residuals
100.00
90.00
R2 = 0.967
Normal Scores
80.00
70.00
60.00
50.00
40.00
30.00
20.00
10.00
0.00
-5.5 -4.75
-4 -3.25 -2.5 -1.75 -1 -0.25 0.5 1.25
2
2.75 3.5 4.25
5
5.75 6.5 7.25
Residuals
Figure 3: Normal scores vs residuals
Residuals vs Fitted Values
8.00
6.00
Residuals
4.00
2.00
0.00
-2.00
-4.00
-6.00
35
45
55
65
75
Fitted Values
Figure 4: Residuals vs fitted values
Residuals vs Battery Life
8
6
Residuals
4
2
0
-2
-4
-6
0
10
20
30
40
50
60
70
80
90
Battery Life
Figure 5: Residuals vs battery life
The 95% confidence interval on the difference between the two treatments was also determined to verify that zero
does not lie within the interval. If zero lies within this interval the conclusion may be made that there is no
difference in means. The equation used to calculate this is [1, 12]:
Yi. – Yj. ± t.025, 12 √(2 * MSE / n)
(17)
It was found with 95% confidence that 38.276 < μ72deg – μ170deg < 44.564.
One other method used to determine if there was a significant difference in sample means was Tukey’s Test. This
test analyzes the differences between means and compares it to a Studentized range statistic “q”. The difference in
means is considered significantly different if [1, 12]
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where
|Yi – Yj| > Tα
(18)
Tα = qα(a, f) √(MSE/n)
(19)
From the data samples and treatment means |Y1 – Y2| was found to be 41.42 and Tα to be q.05(2, 12)√MSE/n = 4.44.
Having demonstrated that the means are quite different, we now move on to estimating the activation energy for the
chemical reaction is estimated. Referring back to equation (5), the values of T1, T2, and R are known, the ratio of
the reaction rate constants, k2/k1 = 77.7 minutes / 36.3 minutes = 2.14. Though the reaction rate is expressed in
more complicated units, those units cancel with the only remaining difference is the difference in times. Equation
(20) solves for Ea.
⎛ − Ea ⎛ 1 1 ⎞ ⎞
⎜ − ⎟⎟
R ⎜⎝ T2 T1 ⎟⎠ ⎟⎠
⎜
⎜
k2
= e⎝
k1
⎛ − Ea ⎛ 1
1 ⎞⎞
⎜⎜
−
⎜
⎟ ⎟⎟
77.7
=
= e⎝ 8.3145 ⎝ 350 295 ⎠ ⎠ Î Ea = 11.9 kJ/mol
36.3
(20)
CONCLUSIONS
An analysis on the life span of dry cell batteries with temperature variability was conducted. The batteries were
subjected to a specific temperature, 72oF and 170oF, for 15 minutes and then run until the light emitted by the
flashlight was determined not visible anymore. Once the tests were completed, the data was analyzed and interpreted
using ANOVA, residuals, confidence interval, and Tukey’s Test to determine if temperature affects the life
expectancy of dry cell batteries.
Analyzing the results from the ANOVA table allows the conclusion to be made that temperature has a significant
effect on the life span of dry cell batteries since F0 is greater than F.05, 1, 12. This finding indicates the rejection of the
original null hypothesis, H0. The residuals found from the treatment means and individual samples were used to plot
figures 3, 4, and 5. These plots indicate that the errors tend to follow a straight line indicating normality; the
variance is constant between errors; and significant difference in mean battery life expectancy between 72oF and
170oF, respectively. The resulting 95% confidence interval found indicated that zero did not lie with in the interval,
indicating that there is a significant difference between the means for battery life at different temperatures. Tukey’s
Test also indicated a significant difference in the means for battery life at different temperatures since |Y72 – Y170|
was greater than Tα. However, Tukey’s Test did not provide any value added information since only two treatment
means were being compared. From these findings, it can be concluded that increased temperature decreases the life
span of dry cell batteries by altering the rate at which the internal chemical reaction takes place; and many spares
would be needed if the equipment being used was subject to high environmental temperatures. Finally, the chemical
reaction’s overall activation energy was calculated using the measured average life spans.
For future research and development, different brands of AA batteries could be tested and analyzed by measuring
voltage, current, or power. The experiments could be run until the battery voltage reaches 5% of the starting voltage.
This may also be applied to larger sized batteries. Studies may also be performed on different types of batteries such
as lead-acid, lithium-ion, and zinc-air batteries as well as based on the user’s environmental and situational needs.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Hines, W., Probability and Statistics in Engineering Fourth Edition, John Wiley & Sons, New York, 2003.
http://antoine.frostburg.edu/chem/senese/101/ kinetics/faq/temperature-and-reaction-rate.shtml
http://electronics.howstuffworks.com/battery3.htm
http://en.wikipedia.org/wiki/Primary_cell
http://gardenofpraise.com/sci998.htm
http://michaelbluejay.com/batteries
http://www.newgenerationbooks.com/AA2/ SampleDC_files/image012.jpg
2009 ASEE Southeast Section Conference
[8]
[9]
[10]
[11]
[12]
http://www.science.uwaterloo.ca/~cchieh/cact/ c123/chmkntcs.html
http://www.statsoft.com/textbook/stanman.html
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Montgomery, D. C., Design and Analysis of Experiments, 6th Edition, John Wiley & Sons, New York, 2005.
R. RADHARAMANAN
Dr. R. Radharamanan is a Professor of Industrial and Systems Engineering in the Department of Mechanical and
Industrial Engineering at Mercer University in Macon, Georgia. He has twenty-eight years of teaching, research,
and consulting experiences. His previous administrative experiences include: President of International Society for
Productivity Enhancement (ISPE), Acting Director of Industrial Engineering as well as Director of Advanced
Manufacturing Center at Marquette University, and Research Director of CAM and Robotics Center at San Diego
State University. His primary research and teaching interests are in the areas of manufacturing systems, quality
engineering, and product and process development. He has organized and chaired three international conferences,
co-chaired two, and organized and chaired one regional seminar. He has received two teaching awards, several
research and service awards in the United States and in Brazil. His professional affiliations include ASEE, IIE,
ASQ, SME, ASME, and ISPE.
JENG-NAN JUANG
Dr. Dr. Juang received the B.S. degree in Electrical Engineering from National Taiwan Ocean University, Keelung,
Taiwan and the M.S. and Ph.D. degrees from Tennessee Technological University, Tennessee. He has been a faculty
member of the Electrical and Computer Engineering at Mercer University since 1987. He organized and chaired the
“International Workshop on Future Aerospace Maintenance & Technology” at Taipei in Taiwan. He also served as a
prime consultant for Fiber Optic Rate Gyroscope Development Program (A grant from Wright-Patterson Air Force
Base, Ohio, and USA). He has received one teaching award, and two research fellow awards from NASA Langley
Research Center, Virginia, USA. His professional affiliations include: ASEE, IEEE, AIAA, NATPA, NSPE,
CAPASUS, and AAUP.
2009 ASEE Southeast Section Conference